Random dense countable sets: characterization by independence

A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are Brownian local minima and unordered infinite sample. They are identically distributed; the former ad hoc proof of this fact is now superseded by a general result. Introduction Random dense countable sets arise naturally from various probabilistic models. Their examination is impeded by the singular nature of the set DCS(0, 1) of all dense countable subsets of (say) the interval (0, 1). This set is not a Polish space, not even a standard Borel space. Nevertheless the idea of random elements of DCS(0, 1) and their distributions can be formalized. An appropriate framework proposed in [2, Sect. 1] is used here. Two examples of random dense countable sets are compared in [2]. One example, ‘Brownian local minima’, is the random set M = {s ∈ (0, 1) : ∃ε > 0 ∀t ∈ (s− ε, s) ∪ (s, s+ ε) Bs < Bt} of local minimizers on (0, 1) of the Brownian motion (Bt)t. The other example, ‘unordered infinite sample’, is the random set S = {U1, U2, . . . } = {s ∈ (0, 1) : ∃n Un = s} where U1, U2, . . . are independent random variables distributed uniformly on (0, 1). The main result of [2] states thatM and S are identically distributed, which means existence of such a joining between the Brownian motion (Bt)t and the sequence (Un)n that M = S a.s. Independence of Brownian increments on disjoint time intervals (a, b) and (c, d) implies independence of ‘fragments’ M ∩ (a, b) and M ∩ (c, d) ofM (see